Is there a way to solve for the orthogonal matrix, $Q$, in
$QMQ^\text{T} = L^\text{T}GL$
where $M$ is a known anti-diagonal matrix and the right hand side is known ($L$ is lower-triangular from a Cholesky decomposition, and $G$ is a Gram matrix)?
For context, the reason I want this is that I'm trying to get a set of vectors from the Gram matrix that have inner products $\langle x, x \rangle = x^\text{T}Mx$ so I thought to try rotate the vectors in $L$ such that this is true.
In general, no. If $G$ is a Gram matrix, $L^TGL$ is positive semidefinite. However, as $M$ is anti-diagonal, if $M$ possesses any nonzero off-diagonal entry, $M$ and $QMQ^T$ cannot possibly be positive semidefinite.